RealClosure TheFieldΒΆ

reclos.spad line 867

This domain implements the real closure of an ordered field.

0: %
from AbelianMonoid
1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, Fraction Integer) -> %
from RightModule Fraction Integer
*: (%, Integer) -> %
from RightModule Integer
*: (%, TheField) -> %
from RightModule TheField
*: (Fraction Integer, %) -> %
from LeftModule Fraction Integer
*: (Integer, %) -> %
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
*: (TheField, %) -> %
from LeftModule TheField
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
/: (%, %) -> %
from Field
<: (%, %) -> Boolean
from PartialOrder
<=: (%, %) -> Boolean
from PartialOrder
=: (%, %) -> Boolean
from BasicType
>: (%, %) -> Boolean
from PartialOrder
>=: (%, %) -> Boolean
from PartialOrder
^: (%, Fraction Integer) -> %
from RadicalCategory
^: (%, Integer) -> %
from DivisionRing
^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
abs: % -> %
from OrderedRing
algebraicOf: (RightOpenIntervalRootCharacterization(%, SparseUnivariatePolynomial %), OutputForm) -> %
algebraicOf(char) is the external number
allRootsOf: Polynomial % -> List %
from RealClosedField
allRootsOf: Polynomial Fraction Integer -> List %
from RealClosedField
allRootsOf: Polynomial Integer -> List %
from RealClosedField
allRootsOf: SparseUnivariatePolynomial % -> List %
from RealClosedField
allRootsOf: SparseUnivariatePolynomial Fraction Integer -> List %
from RealClosedField
allRootsOf: SparseUnivariatePolynomial Integer -> List %
from RealClosedField
annihilate?: (%, %) -> Boolean
from Rng
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
approximate: (%, %) -> Fraction Integer
from RealClosedField
associates?: (%, %) -> Boolean
from EntireRing
associator: (%, %, %) -> %
from NonAssociativeRng
characteristic: () -> NonNegativeInteger
from NonAssociativeRing
coerce: % -> %
from Algebra %
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Fraction Integer -> %
from RetractableTo Fraction Integer
coerce: Integer -> %
from RetractableTo Integer
coerce: TheField -> %
from RetractableTo TheField
commutator: (%, %) -> %
from NonAssociativeRng
divide: (%, %) -> Record(quotient: %, remainder: %)
from EuclideanDomain
euclideanSize: % -> NonNegativeInteger
from EuclideanDomain
expressIdealMember: (List %, %) -> Union(List %, failed)
from PrincipalIdealDomain
exquo: (%, %) -> Union(%, failed)
from EntireRing
extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %)
from EuclideanDomain
extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed)
from EuclideanDomain
factor: % -> Factored %
from UniqueFactorizationDomain
gcd: (%, %) -> %
from GcdDomain
gcd: List % -> %
from GcdDomain
gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %
from GcdDomain
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
inv: % -> %
from DivisionRing
latex: % -> String
from SetCategory
lcm: (%, %) -> %
from GcdDomain
lcm: List % -> %
from GcdDomain
lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %)
from LeftOreRing
leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
mainCharacterization: % -> Union(RightOpenIntervalRootCharacterization(%, SparseUnivariatePolynomial %), failed)
mainCharacterization(x) is the main algebraic quantity of x (SEG)
mainDefiningPolynomial: % -> Union(SparseUnivariatePolynomial %, failed)
from RealClosedField
mainForm: % -> Union(OutputForm, failed)
from RealClosedField
mainValue: % -> Union(SparseUnivariatePolynomial %, failed)
from RealClosedField
max: (%, %) -> %
from OrderedSet
min: (%, %) -> %
from OrderedSet
multiEuclidean: (List %, %) -> Union(List %, failed)
from EuclideanDomain
negative?: % -> Boolean
from OrderedRing
nthRoot: (%, Integer) -> %
from RadicalCategory
one?: % -> Boolean
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
positive?: % -> Boolean
from OrderedRing
prime?: % -> Boolean
from UniqueFactorizationDomain
principalIdeal: List % -> Record(coef: List %, generator: %)
from PrincipalIdealDomain
quo: (%, %) -> %
from EuclideanDomain
recip: % -> Union(%, failed)
from MagmaWithUnit
relativeApprox: (%, %) -> Fraction Integer
relativeApprox(n, p) gives a relative approximation of n that has precision p
rem: (%, %) -> %
from EuclideanDomain
rename!: (%, OutputForm) -> %
from RealClosedField
rename: (%, OutputForm) -> %
from RealClosedField
retract: % -> Fraction Integer
from RetractableTo Fraction Integer
retract: % -> Integer
from RetractableTo Integer
retract: % -> TheField
from RetractableTo TheField
retractIfCan: % -> Union(Fraction Integer, failed)
from RetractableTo Fraction Integer
retractIfCan: % -> Union(Integer, failed)
from RetractableTo Integer
retractIfCan: % -> Union(TheField, failed)
from RetractableTo TheField
rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
rootOf: (SparseUnivariatePolynomial %, PositiveInteger) -> Union(%, failed)
from RealClosedField
rootOf: (SparseUnivariatePolynomial %, PositiveInteger, OutputForm) -> Union(%, failed)
from RealClosedField
sample: %
from AbelianMonoid
sign: % -> Integer
from OrderedRing
sizeLess?: (%, %) -> Boolean
from EuclideanDomain
smaller?: (%, %) -> Boolean
from Comparable
sqrt: % -> %
from RealClosedField
sqrt: (%, PositiveInteger) -> %
from RealClosedField
sqrt: Fraction Integer -> %
from RealClosedField
sqrt: Integer -> %
from RealClosedField
squareFree: % -> Factored %
from UniqueFactorizationDomain
squareFreePart: % -> %
from UniqueFactorizationDomain
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
unit?: % -> Boolean
from EntireRing
unitCanonical: % -> %
from EntireRing
unitNormal: % -> Record(unit: %, canonical: %, associate: %)
from EntireRing
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

Algebra Integer

Algebra TheField

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

BiModule(Integer, Integer)

BiModule(TheField, TheField)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicZero

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable

DivisionRing

EntireRing

EuclideanDomain

Field

FullyRetractableTo Fraction Integer

FullyRetractableTo TheField

GcdDomain

IntegralDomain

LeftModule %

LeftModule Fraction Integer

LeftModule Integer

LeftModule TheField

LeftOreRing

Magma

MagmaWithUnit

Module %

Module Fraction Integer

Module Integer

Module TheField

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

OrderedAbelianGroup

OrderedAbelianMonoid

OrderedAbelianSemiGroup

OrderedCancellationAbelianMonoid

OrderedRing

OrderedSet

PartialOrder

PrincipalIdealDomain

RadicalCategory

RealClosedField

RetractableTo Fraction Integer

RetractableTo Integer

RetractableTo TheField

RightModule %

RightModule Fraction Integer

RightModule Integer

RightModule TheField

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

UniqueFactorizationDomain

unitsKnown